Question

The complete solution of the current in the sinusoidal response of R-L-C circuit is?

(a) i = c1 e^(K1+K2)t + c1 e^(K1-K2)t – V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

(b) i = c1 e^(K1+K2)t + c1 e^(K1-K2)t  – V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ-tan^-1)⁡((1/ωC-ωL)/R))

(c) i = c1 e^(K1+K2)t + c1 e^(K1-K2)t + V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

(d) i = c1 e^(K1+K2)t + c1 e^(K1-K2)t  + V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ-tan^-1)⁡((1/ωC-ωL)/R))

The question was asked in final exam.

My question comes from Sinusoidal Response of an R-L-C Circuit topic in division Transients of Network Theory

Answer 1

Right answer is (c) i = c1 e^(K1+K2)t + c1 e^(K1-K2)t + V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R))

Explanation: The complete solution for the current becomes  i = c1 e^(K1+K2)t + c1 e^(K1-K2)t + V/√(R^2+(1/ωC-ωL)^2) cos⁡(ωt+θ+tan^-1)⁡((1/ωC-ωL)/R)).